In order to show that with the above limits (REF ) we reach all the eigenvalues in the entry (REF ) when \(\eta \equiv \widehat{\eta }\) , namely, that \( \Lambda ^0_m(\widehat{\eta })=\mathbf {\widehat{\Lambda }}^0_m(\widehat{\eta })\) , it suffices to show that the set \(\lbrace {\bf \widehat{U}}^0_m\rbrace _{m=1}^\infty \)  forms a basis of \(L^2(\varpi ^0)\) . Indeed, this is a classical process of contradiction (see, for instance, Section III.1 of [1]}
and Section III.9.1 of [2]}).
In this way, we have proved that (REF ) holds for any \(p\in \mathbb {N}\) , where \(\lbrace {\bf \widehat{\Lambda }}^0_p(\widehat{\eta })\rbrace _{p=1}^\infty \)  are the set of eigenvalues of (REF )–(REF ) when \(\eta \equiv \widehat{\eta }\)  and the eigenfunctions \(\lbrace {\bf \widehat{U}}^0_p \rbrace _{p=1}^\infty \)  form an orthonormal basis of \(L^2(\varpi ^0)\) . Consequently, the sequence (REF ) coincides with (REF ), and the theorem is proved.
